You may talk to a friend, discuss the questions and potential directions for solving them. However, you need to write your own solutions and code separately, and not as a group activity.
Do not write your name on the assignment.
Write your code in the Code cells and your answer in the Markdown cells of the Jupyter notebook. Ensure that the solution is written neatly enough to understand and grade.
Use Quarto to print the .ipynb file as HTML. You will need to open the command prompt, navigate to the directory containing the file, and use the command: quarto render filename.ipynb --to html. Submit the HTML file.
The assignment is worth 100 points, and is due on 1st November 2023 at 11:59 pm.
Some questions in this assignment do not have a single correct answer. As data visualization is subject to interpretation, any logically sound answer / explanation is acceptable.
There is a bonus question worth 20 points. However, there is no partial credit for the bonus question. You will get 20 or 0. If everything is correct, you can score 120 out of 100 in the assignment.
Exploring factors associated with profitability of a movie
In this assignment we’ll attempt to find the factors (or variables) that make a movie profitable.
Read the movies data from here, in a Pandas DataFrame named as movies_data. The profit of movie is defined as:
\[profit = Worldwide \ Gross - Production \ Budget\]
Code
import pandas as pdurl ='https://raw.githubusercontent.com/vega/vega-datasets/master/data/movies.json'movies_data = pd.read_json(url)movies_data.head()
Title
US Gross
Worldwide Gross
US DVD Sales
Production Budget
Release Date
MPAA Rating
Running Time min
Distributor
Source
Major Genre
Creative Type
Director
Rotten Tomatoes Rating
IMDB Rating
IMDB Votes
0
The Land Girls
146083.0
146083.0
NaN
8000000.0
Jun 12 1998
R
NaN
Gramercy
None
None
None
None
NaN
6.1
1071.0
1
First Love, Last Rites
10876.0
10876.0
NaN
300000.0
Aug 07 1998
R
NaN
Strand
None
Drama
None
None
NaN
6.9
207.0
2
I Married a Strange Person
203134.0
203134.0
NaN
250000.0
Aug 28 1998
None
NaN
Lionsgate
None
Comedy
None
None
NaN
6.8
865.0
3
Let's Talk About Sex
373615.0
373615.0
NaN
300000.0
Sep 11 1998
None
NaN
Fine Line
None
Comedy
None
None
13.0
NaN
NaN
4
Slam
1009819.0
1087521.0
NaN
1000000.0
Oct 09 1998
R
NaN
Trimark
Original Screenplay
Drama
Contemporary Fiction
None
62.0
3.4
165.0
Time trend
Let us analyze if the profitability of a movie is associated with the time of its release.
Month of release
Make an appropriate plot to visualize the mean profit of movies released each month.
Hint:
Use the Pandas function to_datetime() to convert Release Date to a datetime datatype.
Use the library datetime to extract the month from Release Date.
(6 points)
Code
import datetimeimport matplotlib.pyplot as pltmovies_data = movies_data.dropna(subset=['Release Date', 'Production Budget', 'Worldwide Gross'])movies_data['Release Date'] = pd.to_datetime(movies_data['Release Date'], errors='coerce') #First convert the Release Date to Datetime data typemovies_data['Profit'] = movies_data['Worldwide Gross'] - movies_data['Production Budget']movies_data['Month'] = movies_data['Release Date'].dt.month #Extract month from Release Datemean_profit = movies_data.groupby('Month')['Profit'].mean()mean_profit.plot(kind='bar', figsize=(10, 6))plt.title('Mean Profit of Movies Released Each Month')plt.xlabel('Month')plt.ylabel('Mean Profit')plt.xticks(ticks=range(12), labels=['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'])plt.show()
Based on the plot, which seasons have been the most and least profitable (on an average) for a movie release. Don’t worry about the exact start and end date of seasons. Don’t perform any computations. Just make comments based on the plot. You can use seasons such as early summer, late spring etc.
(2 points)
The Most Profitable seasons are May to Jul and Nov to Dec .
Movies released during the first period (May to July tend to have higher average profits. This could be due to the summer break when more people have free time to watch movies. The cozy weather also encourages more gatherings. Watching movies is always a good choice for leisure with friends.
Movies released during the second period (November to December) is the holiday season. Families are more likely to watch movies together to celebrate.
The Least Profitable seasons are Late Winter(Jan to Feb), Late Spring(Apr), and Fall (Sep to Oct) season.
There are fewer major holidays and less free time for potential viewers during these seasons. The fall season, in particular, is the beginning of an academic year, so teenagers and young adults, who are the major viewer groups. have few time for leisure. Consequently, the seasons are less profitable due to a decline in the number of viewers.
Month of release with number of movies in each genre
Now that we know the most profitable season for releasing movies, let us visualize if some genres are more popular during certain seasons.
Use the code below to create a new column called genre.
Code
#Combining Major Genremovies_data['genre'] = movies_data['Major Genre'].apply(lambda x:'Comedy'if x!=Noneand'Comedy'in x else'Horror'if x!=Noneand'Thriller'in x else'Action/Adventure'if x!=Noneand ('Action'in x or'Adventure'in x) else'Musical/Western'if x!=Noneand ('Musical'in x or'Western'in x or'Concert'in x) else x)
Make an appropriate plot to visualize the number of movies released for each genre in each calendar month.
(8 points)
Code
genre_month_counts = movies_data.groupby(['genre', 'Month']).size().unstack()genre_month_counts.transpose().plot(kind='bar', stacked=True, figsize=(12, 6))plt.title('Number of Movies Released Each Month by Genre')plt.xlabel('Month')plt.ylabel('Number of Movies')plt.xticks(ticks=range(12), labels=['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'])plt.legend(title='Genre')plt.show()
Based on the above plot, which genre is the most popular during the most profitable season of release? And which genre is the most popular during the least profitable season of release?
(2 points)
Based on the plot, the most popular genre during the first profitable season is Action/Adventure followed by Comedy. Because the major viewer group of the season is likely to be adolescents and young adults. These two types appeal to them. While during the second profitable season, Drama and Comedy gain popularity, because during the holiday seasons, the viewer groups are more heterogenous. People watch movies with their families, so the two genre have the largest range of potential audiences.
Month of release with proportion of movies in each genre
Visualize the proportion of movies in each genre for each month of release.
Use the code below to re-arrange your data that will help with creating the visualization
Make a 100% stacked barplot with the Pandas plot() function
Use the argument bbox_to_anchor with the Matplotlib function legend() to place the legend outside the plot area.
(8 points)
Code
genre_proportion_release_month = pd.crosstab(index=movies_data['Month'], columns=movies_data['genre'], normalize="index")# Plottingax = genre_proportion_release_month.plot(kind='bar', stacked=True, figsize=(12, 6), width=0.8, colormap='tab20')plt.title('Proportion of Movies in Each Genre by Release Month')plt.xlabel('Month')plt.ylabel('Proportion')plt.xticks(ticks=range(12), labels=['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'], rotation=0)plt.legend(title='Genre', bbox_to_anchor=(1.05, 1), loc='upper left')plt.tight_layout()plt.show()
Which genre is the most popular during the month of May, and which one is the most popular during December?
(2 points)
The most popular one in May is Action/Adventure. The most popular one in December is Drama
Year of release with genre
Make an appropriate figure to visualize the average profit of movies of each genre for each year. Consider only the movies released from 1991 to 2010. Also show the 95% confidence interval in the average profit.
Hint:
Use the library datetime to extract year from Release Date.
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
/Users/jinwens/anaconda3/lib/python3.11/site-packages/seaborn/axisgrid.py:848: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
func(*plot_args, **plot_kwargs)
Based on the figure above, which genre’s profitability seems to be increasing over the years, and which genre has the least uncertainty in profit for most of the years.
(2 points)
Horror seems to be increasing over the years. Actions/Adventure has the least uncertainty in profit for most of the years.
/var/folders/vh/bnk8gjpn5998vq44hzdttvwc0000gn/T/ipykernel_2593/2851320192.py:1: FutureWarning: The default value of numeric_only in DataFrame.corr is deprecated. In a future version, it will default to False. Select only valid columns or specify the value of numeric_only to silence this warning.
sns.heatmap(movies_data.corr())
Associations
Pairplot / heatmap
Make a pairplot and heatmap of all the continuous variables in the data.
# Some category like IMDB voting and Running Time can be modified and used as categorical variable. # Without further information, I sorted out the the following variables which are always continuous for my answer. continuous_vars = movies_data[['Production Budget', 'US Gross', 'Worldwide Gross', 'Profit', 'IMDB Rating', 'Rotten Tomatoes Rating', 'MPAA Rating', 'US DVD Sales']]continuous_vars = continuous_vars.dropna()sns.pairplot(continuous_vars)plt.suptitle('Pairplot of Continuous Variables', y=1.02)plt.show()corr_matrix = continuous_vars.corr()sns.heatmap(corr_matrix, annot=True, fmt=".2f")plt.title('Heatmap of Continuous Variables')plt.show()
Are there any trends that you can see in the pairplot, but not in the heatmap?
(2 points)
Pairplot visually displays relationships between pairs of continuous variables, allowing us to see trends, patterns, and outliers in scatter plots. Heatmap visually displays the correlation coefficients between pairs of continuous variables, allowing us to see the strength and direction of linear relationships.
The pair plot reveals the concentration of different observations, while the heatmap only shows the coefficient. If some samples spread widely across the distribution, we can see a very loose relationship between the independent and dependent variables through the scatter plot. For example, the left-most graph shows that most observations are concentrated in the left bottom section. We can not really see a convincing/obvious positive relationship looking at it, but the heat map suggests that the production budget has a positive correlation with all the other three.
Based on the plots in 2(a)(i), which variables are associated with profit?
(2 points)
US Gross, US DVD Sales, and World Gross are positively associated with profit. They are variables that show a clear trend or pattern in relation to profit in the pairplot and have a high absolute correlation coefficient with profit in the heatmap.
Among the variables listed in 2(a)(iii), select a subset of variables such that none of them are highly associated with each other. The rest of the variables identified in 2(a)(iii) are redundant with regard to association with profit.
(2 points)
U.S. Gross is not highly associated with the Production Budget. It is highly associated with Profit. Worldwide Gross is not highly associated with the Production Budget. It is highly associated with Profit. Profit is not highly associated with the Production Budget. Production Budget is redundant to determine the Profit.
Make an appropriate figure to visualize the association of the number of IMDB votes with profit for each genre (use the variable genre). Which genre has the highest association between profit and IMDB votes?
Documentary shows a clear upward trend(the most steep slope), meaning that as the number of IMDB votes increases, the profit also tends to increase at a comparatively more salient scale.
Make an appropriate figure to visualize the association between the number of IMDB votes and profit, for each combination of the fiction type (use the variable fiction) and the movie rating (use the variable rating).
For which combination of fiction and rating categories do you observe the highest association between IMDB votes and profit?
(8 points)
Hint: Use row and col attributes of the Seaborn Facetgrid() object.
fiction rating
Contemporary Not R rated 0.644792
R rated 0.489156
other Not R rated 0.640152
R rated 0.582098
dtype: float64
Contemporary Fiction- Not R rated has the highest association between IMDB votes and profit.
Profit based on movie director
Consider the directors who have directed more than 10 movies (based on the dataset). Make a horizontal barplot that shows the mean profit of the movies of these directors along with the 95% confidence interval. Sort the bars of the barplot such that the director with the highest mean profit is at the top.
If the dataset director_with_more_than_10_movies has only those movies that correspond to directors with more than 10 movies, then the following code will give you the order in which the names of the directors must appear in the barplot:
(8 points)
Code
director_counts = movies_data['Director'].value_counts()directors_more_than_10 = director_counts[director_counts >10].indexdirector_with_more_than_10_movies = movies_data[movies_data['Director'].isin(directors_more_than_10)]# Creating the barplotplt.figure(figsize=(10, 8))sns.barplot( data=director_with_more_than_10_movies, y='Director', x='Profit', order=director_with_more_than_10_movies[['Director','Profit']].groupby('Director').mean().sort_values(by='Profit', ascending=False).index, ci=95)plt.xlabel('Mean Profit')plt.ylabel('Director')plt.title('Mean Profit of Directors with more than 10 Movies with 95% Confidence Interval')plt.show()
/var/folders/vh/bnk8gjpn5998vq44hzdttvwc0000gn/T/ipykernel_48097/2425434583.py:8: FutureWarning:
The `ci` parameter is deprecated. Use `errorbar=('ci', 95)` for the same effect.
sns.barplot(
Based on the above plot, which director has the highest mean profitability, and which one has the highest variation in profitability?
(2 points)
Steven Spielberg has the highest mean profit. Sam Raimi has the widest confidence interval, so he has the highest variation in profitability.
Distributions
Distribution of profit based on genre (boxplots)
Make boxplots to visualize the distribution of profit based on genre. Based on the plot, which genre has the most profitable movies?
(6 points)
Code
# D.3.1.1 Boxplotsplt.figure(figsize=(10, 8))sns.boxplot(data=movies_data, x='Profit', y='genre')plt.title('Distribution of Profit Based on Genre (Boxplots)')plt.xlabel('Profit')plt.ylabel('genre')plt.show()
Action/Adventure has the most profitable movie.
Which genre has the most variation in profit, and which one has the least?
(2 points)
Action/Adventure has the most variation in profit. Documentary has the least variation.
Distribution of profit based on genre (density plots)
Make density plots of profit based on genre. Adjust the limit on the horizonal axis, so that the plots are clearly visible.
(6 points)
Code
plt.figure(figsize=(10, 8))for genre in movies_data['genre'].unique():if pd.notna(genre): sns.kdeplot(movies_data[movies_data['genre'] == genre]['Profit'], label=genre, fill=True)plt.title('Distribution of Profit Based on Genre (Density Plots)')plt.xlabel('Profit')plt.ylabel('Density')plt.xlim(-100000000, 500000000) # Adjust the limit on the horizontal axisplt.legend()plt.show()
What additional insight / trend can you seen in the above plot that you cannot see with the boxplots?
(2 points)
The density plot clearly shows the mode(s) of the distribution. If a genre has multiple peaks （like documentary), it indicates that there are multiple common profit values for movies in that genre. The density plot can visually represent the skewness of the distribution. It shows whether the profit distribution of a genre is symmetric, right-skewed (long tail towards higher profits), or left-skewed (long tail towards lower profits). In the graph, all distributions are left- skewed, which is not previously reviewed by the boxplot.
Insights
From all the visualizations above, describe the insights you get about the factors associated with the profitability of a movie.
Also, elaborate on the extent to which these trends can be generalized. For example, comment on whether these trends be generalized to the current time and all the Hollywood movies? If not, is there any time period or type of movie to which these trends can be applicable?
(4+ 4 points)
Upon analyzing various visualizations derived from the movie dataset, several insightful trends emerge regarding the factors associated with a movie’s profitability. The profitability of the action/ adventure genre is leading the industry across different years. Notably, [genre, director, or IMDB ratings] appear to have a substantial impact, demonstrating [average profits genres. Furthermore, the influence of release season is also evident, revealing that movies that are out in summer and the holiday seasons have higher box office returns. A strong IMDB rating does not necessarily correlate with higher profitability. Although we see a positive correlation， the magnitude of the coefficient is surprisingly low.
Regarding the generalization of these trends, caution is advised. The cinematic landscape can be constantly evolving, potentially limiting the applicability of these insights to the current era. Moreover, the targeted audiences of the same genre can be different depending on the age group, the trends may not universally apply across diverse movie categories and global cinemas. Thus, while the observed patterns offer valuable perspectives, their broader applicability may be constrained by temporal, geographical, and cinematic variations.
Bonus question: Stock Market
This question is worth 20 points. However, there is no partial credit. It will be 20 or 0.
The stock market is made up of exchanges, such as the New York Stock Exchange and the Nasdaq. Stocks are listed on a specific exchange, which brings buyers and sellers together and acts as a market for the shares of those stocks. The exchange tracks the supply and demand — and directly related, the price — of each stock.
A market index tracks the performance of a group of stocks, which either represents the market as a whole or a specific sector of the market, like technology or retail companies. You’re likely to hear most about the S&P 500, the Nasdaq composite and the Dow Jones Industrial Average; they are often used as proxies for the performance of the overall market.
There are two types of investment stratigies: active and passive.
Active investing
An active investment strategy involves using the information acquired by expert stock analysts to actively buy and sell stocks with specific characteristics. The goal is to beat the results of the indices and general stock market with higher returns and/or lower risk.
Passive Investing
Passive investors have a buy-and-hold mentality that focuses on benefitting from the overall increase in market prices over time. One of the major benefits of passive investing is that it minimizes the mistakes investors can make when they react emotionally to every move of the stock market.
The easiest way to implement a passive approach is to buy and hold an index fund that follows one of the major indices like the S&P 500, Dow Jones, or Russell 2000 (small-cap stocks). These funds pool money from multiple investors to buy the individual stocks, bonds, or securities that make up their market index. When the index changes its components, the index funds that follow it also switch up their holdings to match.
Tasks
In this exercise, we use S&P 500 index as an example to explore the gains/returns for passive invesement.
Return
The data is provided in a csv file, with dates roughly from 2000 to 2022. And we use the column Close as the price on a specific day.
If we buy the index on \(t_1\) and sell it on \(t_2\), the return is defined as \[r_{t_1,t_2} = (P_{t_2}-P_{t_1})/P_{t_1}\].
Sometimes we are interested on the return on holding the index for a specific period \(T\), the return is \[r_{t,T} = (P_{t+T}-P_{t})/P_{t}\] where \(t\) is the date we buy the stock and \(T\) is the holding period. Since the stock is not traded on every day, when calculating \({t+T}\) we simply skip the non-trading dates.
Risk
If we take the return as a random variable, we could use standard deviation as its risk. A risk-averse investor expects a stable (low volatility) return.
Sharpe Ratio
People like high return and low risk investment. But on the other hand, in the market high return always associates high risk (e.g. stock) and low risk means low return (e.g. treasury bonds).
The Sharpe ratio compares the excess return of an investment with its risk to make a single measure: \[SR=(R-R_f)/\sigma\] where \(R\) and \(\sigma\) are the expected return and stddev for the investment and \(R_f\) is the risk free rate.
Risk free rate is the rate of return offered by an investment that carries zero risk. In reality there is no truly risk free rate, but we usually take something like three-month U.S. Treasury bill as a proxy as risk free rate. To be simple, in this exercise we just take risk free rate as 0.
Read the data from sp500.csv. The dataset can be found in the Datasets section of the book.
Return on investment
Impatient investor (daily)
Suppose there is an investor who only holds the index for a single day (buy yesterday sell today).
Based on the data,
Show the histogram graph for all the possible returns.
What is the expected return, risk and sharpe ratio?
Is the return significantly greater than zero (a.k.a positive return) (use a threshold 0.01 for \(p\)-value) ?
HINT: use scipy.stats.ttest_1samp to do one-sided mean test. (We ignore the fact that T-test requires the data are sampled from a population of normal distribution, which might not be true in this exercise)
(6 points)
D.5.1.1 Show the histogram graph for all the possible returns.
D.5.1.2 What is the expected return, risk and sharpe ratio?
D.5.1.3 Is the return significantly greater than zero (a.k.a positive return) (use a threshold 0.01 for -value) ?
Code
import pandas as pdimport matplotlib.pyplot as pltimport numpy as np# 5.1# Load the datasetdata = pd.read_csv('/Users/jinwens/Downloads/sp500.csv')data.head()
Date
Open
High
Low
Close
0
2021-08-24
4484.40
4492.81
4482.28
4486.23
1
2016-07-14
2157.88
2168.99
2157.88
2163.75
2
2001-10-02
1051.33
1051.33
1034.47
1051.33
3
2010-11-11
1213.04
1215.45
1204.49
1213.54
4
2014-06-10
1950.34
1950.86
1944.64
1950.79
Code
data['Date'] = pd.to_datetime(data['Date'])data = data.sort_values('Date')# Remove extra spaces in column namesdata.columns = data.columns.str.strip()# Daily Returndata['Return'] = data['Close'].pct_change()# Drop rows with NaN values in 'Return' column (first row)data = data.dropna(subset=['Return'])# Plot histogram of the daily returnsplt.hist(data['Return'], bins=50, edgecolor='black', alpha=0.7)plt.title('Histogram of Daily Returns')plt.xlabel('Return')plt.ylabel('Frequency')plt.grid(True)plt.show()
# 5.1.3from scipy.stats import ttest_1samp# Perform one-sample t-testt_stat, p_value_two_tailed = ttest_1samp(data['Return'], 0)# One-tailed test, we need to divide the p-value by 2p_value_one_tailed = p_value_two_tailed /2# Check if the return is significantly greater than zerosignificant_return = p_value_one_tailed <0.01t_stat, p_value_one_tailed, significant_return
(1.4710273319906229, 0.07066997605887224, False)
T-Statistic = 1.471
One-Tailed P-Value = 0.071
Significant Positive Return: No, the return is not significantly greater than zero at a 0.01 threshold for the p-value.
Patient Investor (yearly)
Suppose there is an investor who will hold the index for a year (suppose there are 250 trading days in a year). Do the same analysis as the above:
Show the histogram graph for all the possible returns.
What is the expected return, risk and sharpe ratio?
Is the return significantly greater than zero (a.k.a positive return) (use a threshold 0.01 for \(p\)-value) ?
(6 points)
Code
# 5.2.1# Calculate yearly returnsholding_period =250data['Yearly_Return'] = data['Close'].pct_change(periods=holding_period)data_yearly = data.dropna(subset=['Yearly_Return'])# Plot histogram of the yearly returnsplt.hist(data_yearly['Yearly_Return'], bins=50, edgecolor='black', alpha=0.7)plt.title('Histogram of Yearly Returns')plt.xlabel('Return')plt.ylabel('Frequency')plt.grid(True)plt.show()
# 5.2.3t_stat_yearly, p_value_two_tailed_yearly = ttest_1samp(data_yearly['Yearly_Return'], 0)# Since we are doing a one-tailed test, we need to divide the p-value by 2p_value_one_tailed_yearly = p_value_two_tailed_yearly /2# Check if the yearly return is significantly greater than zerosignificant_return_yearly = p_value_one_tailed_yearly <0.01t_stat_yearly, p_value_one_tailed_yearly, significant_return_yearly
(30.7390794360697, 1.6288154028590484e-191, True)
T-Statistic = 30.7391
One-Tailed P-Value = 1.6288
Significant Positive Return: Yes, the return is significantly greater than zero at a 0.01 threshold for the p-value.
From daily to yearly
Explore how the expected return/risk/shape ratio change as we increase our holding period from 1 day to 1 year(250 days).
Show/answer:
At least how many days do you need to hold the index in order to make a significant positive return (threshold 0.01)?
How are the returns associated with the risks for different investment strategies?
Make a graph as shown below.
(18 points)
Code
def is_significant_return(returns, alpha=0.01): t_stat, p_value_two_tailed = ttest_1samp(returns, 0) p_value_one_tailed = p_value_two_tailed /2return p_value_one_tailed < alpha# Finding the minimum holding period for significant positive returnmin_holding_period =Nonefor period inrange(1, holding_period+1): data['Return_Period'] = data['Close'].pct_change(periods=period) data_period = data.dropna(subset=['Return_Period'])if is_significant_return(data_period['Return_Period']): min_holding_period = periodbreakmin_holding_period
3
5.3.1 The index must be held for at least 3 days to achieve a significant positive return at a threshold of 0.01.
5.3.2 I’ll calculate the expected returns and risks for holding periods ranging from 1 day to 250 days and then analyze the relationship between them.
Code
# Lists to store the calculated valuesholding_periods = []expected_returns = []risks = []sharpe_ratios = []# Calculate expected return, risk, and Sharpe ratio for different holding periodsfor period inrange(1, holding_period+1): holding_periods.append(period) data['Return_Period'] = data['Close'].pct_change(periods=period) data_period = data.dropna(subset=['Return_Period']) expected_return_period = data_period['Return_Period'].mean() risk_period = data_period['Return_Period'].std() sharpe_ratio_period = expected_return_period / risk_period if risk_period !=0elseNone expected_returns.append(expected_return_period) risks.append(risk_period) sharpe_ratios.append(sharpe_ratio_period)# Plottingplt.figure(figsize=(10, 6))plt.subplot(3, 1, 1)plt.plot(holding_periods, expected_returns, label='Expected Return', color='blue')plt.title('Expected Return vs. Holding Period')plt.xlabel('Holding Period (days)')plt.ylabel('Expected Return')plt.grid(True)plt.subplot(3, 1, 2)plt.plot(holding_periods, risks, label='Risk', color='red')plt.title('Risk vs. Holding Period')plt.xlabel('Holding Period (days)')plt.ylabel('Risk (Standard Deviation)')plt.grid(True)plt.subplot(3, 1, 3)plt.plot(holding_periods, sharpe_ratios, label='Sharpe Ratio', color='green')plt.title('Sharpe Ratio vs. Holding Period')plt.xlabel('Holding Period (days)')plt.ylabel('Sharpe Ratio')plt.grid(True)plt.tight_layout()plt.show()
5.3.3Expected Return: The expected return generally increases as the holding period increases. However, there are fluctuations, and the return doesn’t always increase with a longer holding period.
Therefore, a holding period of at least 3 days is necessary to achieve a significant positive return.
Risk: The risk, measured as the standard deviation of returns, also generally increases with the holding period, reflecting the increased uncertainty associated with longer investment horizons.
The relationship between return and risk varies, and a longer holding period doesn’t always lead to better risk-adjusted returns.
Sharpe Ratio: The Sharpe ratio, which adjusts returns for risk, tends to fluctuate. For some periods, the increase in expected return is not enough to compensate for the increased risk, leading to a lower Sharpe ratio. However, in some cases, the Sharpe ratio improves with a longer holding period, indicating a better risk-adjusted return.
The relationship between return and risk varies, and a longer holding period doesn’t always lead to better risk-adjusted returns. Evaluating the expected return, risk, and Sharpe ratio can help investors choose a suitable holding period based on their risk tolerance and investment objectives.
